An Operational Environment for Quantum Self-Testing

Matthias Christandl, Nicholas Gauguin Houghton-Larsen, and Laura Mancinska

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2200 Copenhagen, Denmark

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Observed quantum correlations are known to determine in certain cases the underlying quantum state and measurements. This phenomenon is known as $\textit{(quantum) self-testing}$.
Self-testing constitutes a significant research area with practical and theoretical ramifications for quantum information theory. But since its conception two decades ago by Mayers and Yao, the common way to rigorously formulate self-testing has been in terms of operator-algebraic identities, and this formulation lacks an operational interpretation. In particular, it is unclear how to formulate self-testing in other physical theories, in formulations of quantum theory not referring to operator-algebra, or in scenarios causally different from the standard one.
In this paper, we explain how to understand quantum self-testing operationally, in terms of causally structured dilations of the input-output channel encoding the correlations. These dilations model side-information which leaks to an environment according to a specific schedule, and we show how self-testing concerns the relative strength between such scheduled leaks of information. As such, the title of our paper has double meaning: we recast conventional quantum self-testing in terms of information-leaks to an environment – and this realises quantum self-testing as a special case within the surroundings of a general operational framework.
Our new approach to quantum self-testing not only supplies an operational understanding apt for various generalisations, but also resolves some unexplained aspects of the existing definition, naturally suggests a distance measure suitable for robust self-testing, and points towards self-testing as a modular concept in a larger, cryptographic perspective.

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Cited by

[1] Connor Paddock, William Slofstra, Yuming Zhao, and Yangchen Zhou, "An Operator-Algebraic Formulation of Self-testing", Annales Henri Poincaré (2023).

[2] Arthur J. Parzygnat, "Reversing information flow: retrodiction in semicartesian categories", arXiv:2401.17447, (2024).

[3] Pedro Baptista, Ranyiliu Chen, Jędrzej Kaniewski, David Rasmussen Lolck, Laura Mančinska, Thor Gabelgaard Nielsen, and Simon Schmidt, "A mathematical foundation for self-testing: Lifting common assumptions", arXiv:2310.12662, (2023).

[4] Ernest Y. -Z. Tan, "Prospects for device-independent quantum key distribution", arXiv:2111.11769, (2021).

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